Chapter 3: Pressure and Fluid Statics

1. Chapter 3: Pressure and Fluid Statics Eric G. Paterson Department of Mechanical and Nuclear Engineering The Pennsylvania State University Spring 2005

2. Note to Instructors These slides were developed1 during the spring semester 2005, as a teaching aid for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of Mechanical and Nuclear Engineering at Penn State University. This course had two sections, one taught by myself and one taught by Prof. John Cimbala. While we gave common homework and exams, we independently developed lecture notes. This was also the first semester that Fluid Mechanics: Fundamentals and Applications was used at PSU. My section had 93 students and was held in a classroom with a computer, projector, and blackboard. While slides have been developed for each chapter of Fluid Mechanics: Fundamentals and Applications, I used a combination of blackboard and electronic presentation. In the student evaluations of my course, there were both positive and negative comments on the use of electronic presentation. Therefore, these slides should only be integrated into your lectures with careful consideration of your teaching style and course objectives. Eric Paterson Penn State, University Park August 2005 1 These slides were originally prepared using the LaTeX typesetting system (http://www.tug.org/) and the beamer class (http://latex-beamer.sourceforge.net/), but were translated to PowerPoint for wider dissemination by McGraw-Hill.

3. Pressure • Pressure is defined as a normal force exerted by a fluid per unit area. • Units of pressure are N/m2, which is called a pascal (Pa). • Since the unit Pa is too small for pressures encountered in practice, kilopascal (1 kPa = 103 Pa) and megapascal (1 MPa = 106 Pa) are commonly used. • Other units include bar, atm, kgf/cm2, lbf/in2=psi.

4. Absolute, gage, and vacuum pressures • Actual pressure at a give point is called the absolute pressure. • Most pressure-measuring devices are calibrated to read zero in the atmosphere, and therefore indicate gage pressure, Pgage=Pabs - Patm. • Pressure below atmospheric pressure are called vacuum pressure, Pvac=Patm - Pabs.

5. Absolute, gage, and vacuum pressures

6. Pressure at a Point • Pressure at any point in a fluid is the same in all directions. • Pressure has a magnitude, but not a specific direction, and thus it is a scalar quantity.

7. Variation of Pressure with Depth • In the presence of a gravitational field, pressure increases with depth because more fluid rests on deeper layers. • To obtain a relation for the variation of pressure with depth, consider rectangular element • Force balance in z-direction gives • Dividing by Dx and rearranging gives

8. Variation of Pressure with Depth • Pressure in a fluid at rest is independent of the shape of the container. • Pressure is the same at all points on a horizontal plane in a given fluid.

9. Scuba Diving and Hydrostatic Pressure

10. Scuba Diving and Hydrostatic Pressure • Pressure on diver at 100 ft? • Danger of emergency ascent? 1 100 ft 2 Boyle’s law If you hold your breath on ascent, your lung volume would increase by a factor of 4, which would result in embolism and/or death.

11. Pascal’s Law • Pressure applied to a confined fluid increases the pressure throughout by the same amount. • In picture, pistons are at same height: • Ratio A2/A1 is called ideal mechanical advantage

12. The Manometer • An elevation change of Dz in a fluid at rest corresponds to DP/rg. • A device based on this is called a manometer. • A manometer consists of a U-tube containing one or more fluids such as mercury, water, alcohol, or oil. • Heavy fluids such as mercury are used if large pressure differences are anticipated.

13. Mutlifluid Manometer • For multi-fluid systems • Pressure change across a fluid column of height h is DP = rgh. • Pressure increases downward, and decreases upward. • Two points at the same elevation in a continuous fluid are at the same pressure. • Pressure can be determined by adding and subtracting rgh terms.

14. Measuring Pressure Drops • Manometers are well--suited to measure pressure drops across valves, pipes, heat exchangers, etc. • Relation for pressure drop P1-P2 is obtained by starting at point 1 and adding or subtracting rgh terms until we reach point 2. • If fluid in pipe is a gas, r2>>r1 and P1-P2= rgh

15. The Barometer • Atmospheric pressure is measured by a device called a barometer; thus, atmospheric pressure is often referred to as the barometric pressure. • PC can be taken to be zero since there is only Hg vapor above point C, and it is very low relative to Patm. • Change in atmospheric pressure due to elevation has many effects: Cooking, nose bleeds, engine performance, aircraft performance.

16. Fluid Statics • Fluid Statics deals with problems associated with fluids at rest. • In fluid statics, there is no relative motion between adjacent fluid layers. • Therefore, there is no shear stress in the fluid trying to deform it. • The only stress in fluid statics is normal stress • Normal stress is due to pressure • Variation of pressure is due only to the weight of the fluid → fluid statics is only relevant in presence of gravity fields. • Applications: Floating or submerged bodies, water dams and gates, liquid storage tanks, etc.

17. Hoover Dam

18. Hoover Dam

19. Hoover Dam • Example of elevation head z converted to velocity head V2/2g. We'll discuss this in more detail in Chapter 5 (Bernoulli equation).

20. Hydrostatic Forces on Plane Surfaces • On a plane surface, the hydrostatic forces form a system of parallel forces • For many applications, magnitude and location of application, which is called center of pressure, must be determined. • Atmospheric pressure Patm can be neglected when it acts on both sides of the surface.

21. Resultant Force The magnitude of FRacting on a plane surface of a completely submerged plate in a homogenous fluid is equal to the product of the pressure PC at the centroid of the surface and the area A of the surface

22. Center of Pressure • Line of action of resultant force FR=PCA does not pass through the centroid of the surface. In general, it lies underneath where the pressure is higher. • Vertical location of Center of Pressure is determined by equation the moment of the resultant force to the moment of the distributed pressure force. • $Ixx,C is tabulated for simple geometries.

23. Hydrostatic Forces on Curved Surfaces • FR on a curved surface is more involved since it requires integration of the pressure forces that change direction along the surface. • Easiest approach: determine horizontal and vertical components FH and FV separately.

24. Hydrostatic Forces on Curved Surfaces • Horizontal force component on curved surface: FH=Fx. Line of action on vertical plane gives y coordinate of center of pressure on curved surface. • Vertical force component on curved surface: FV=Fy+W, where W is the weight of the liquid in the enclosed block W=rgV. x coordinate of the center of pressure is a combination of line of action on horizontal plane (centroid of area) and line of action through volume (centroid of volume). • Magnitude of force FR=(FH2+FV2)1/2 • Angle of force is a = tan-1(FV/FH)

25. Buoyancy and Stability • Buoyancy is due to the fluid displaced by a body. FB=rfgV. • Archimedes principal : The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upward through the centroid of the displaced volume.

26. Buoyancy and Stability • Buoyancy force FB is equal only to the displaced volume rfgVdisplaced. • Three scenarios possible • rbody<rfluid: Floating body • rbody=rfluid: Neutrally buoyant • rbody>rfluid: Sinking body

27. Example: Galilean Thermometer • Galileo's thermometer is made of a sealed glass cylinder containing a clear liquid. • Suspended in the liquid are a number of weights, which are sealed glass containers with colored liquid for an attractive effect. • As the liquid changes temperature it changes density and the suspended weights rise and fall to stay at the position where their density is equal to that of the surrounding liquid. • If the weights differ by a very small amount and ordered such that the least dense is at the top and most dense at the bottom they can form a temperature scale.

28. Example: Floating Drydock Submarine undergoing repair work on board the AFDM-10 Auxiliary Floating Dry Dock Resolute(AFDM-10) partially submerged Using buoyancy, a submarine with a displacement of 6,000 tons can be lifted!

29. Example: Submarine Buoyancy and Ballast • Submarines use both static and dynamic depth control. Static control uses ballast tanks between the pressure hull and the outer hull. Dynamic control uses the bow and stern planes to generate trim forces.

30. Example: Submarine Buoyancy and Ballast SSN 711 nose down after accidentwhich damaged fore ballast tanks Normal surface trim

31. Example: Submarine Buoyancy and Ballast Damage to SSN 711 (USS San Francisco) after running aground on 8 January 2005.

32. Example: Submarine Buoyancy and Ballast Ballast Control Panel: Important station for controlling depth of submarine

33. Stability of Immersed Bodies • Rotational stability of immersed bodies depends upon relative location of center of gravityG and center of buoyancyB. • G below B: stable • G above B: unstable • G coincides with B: neutrally stable.

34. Stability of Floating Bodies • If body is bottom heavy (G lower than B), it is always stable. • Floating bodies can be stable when G is higher than B due to shift in location of center buoyancy and creation of restoring moment. • Measure of stability is the metacentric height GM. If GM>1, ship is stable.

35. Rigid-Body Motion • There are special cases where a body of fluid can undergo rigid-body motion: linear acceleration, and rotation of a cylindrical container. • In these cases, no shear is developed. • Newton's 2nd law of motion can be used to derive an equation of motion for a fluid that acts as a rigid body • In Cartesian coordinates:

36. Linear Acceleration Container is moving on a straight path Total differential of P Pressure difference between 2 points Find the rise by selecting 2 points on free surface P2 = P1

37. Rotation in a Cylindrical Container Container is rotating about the z-axis Total differential of P On an isobar, dP = 0 Equation of the free surface

38. Examples of Archimedes Principle

39. The Golden Crown of Hiero II, King of Syracuse • Archimedes, 287-212 B.C. • Hiero, 306-215 B.C. • Hiero learned of a rumor where the goldsmith replaced some of the gold in his crown with silver. Hiero asked Archimedes to determine whether the crown was pure gold. • Archimedes had to develop a nondestructive testing method

40. The Golden Crown of Hiero II, King of Syracuse • The weight of the crown and nugget are the same in air: Wc = rcVc = Wn = rnVn. • If the crown is pure gold, rc=rn which means that the volumes must be the same, Vc=Vn. • In water, the buoyancy force is B=rH2OV. • If the scale becomes unbalanced, this implies that the Vc≠ Vn, which in turn means that the rc ≠rn • Goldsmith was shown to be a fraud!

41. Hydrostatic Bodyfat Testing • What is the best way to measure body fat? • Hydrostatic Bodyfat Testing using Archimedes Principle! • Process • Measure body weight W=rbodyV • Get in tank, expel all air, and measure apparent weight Wa • Buoyancy force B = W-Wa = rH2OV. This permits computation of body volume. • Body density can be computed rbody=W/V. • Body fat can be computed from formulas.

42. Hydrostatic Bodyfat Testing in Air? • Same methodology as Hydrostatic testing in water. • What are the ramifications of using air? • Density of air is 1/1000th of water. • Temperature dependence of air. • Measurement of small volumes. • Used by NCAA Wrestling (there is a BodPod on PSU campus).